Cardinality and equivalence relations greg hjorth university of. Measures are important not only because of their intrinsic geometrical and probabilistic signi. Geyer february 26, 2020 1 old probability theory and new all of probability theory can be divided into two parts. Measure theory graduate texts in mathematics 1994th edition by j. There are many other textbooks on measure theory that cover most or. Given the short time that we have to spend on this, we are really only going to be able to introduce the relevant concepts, and try to give an idea of why they are. Volume 2 aims to lay out some of the fundamental results of pure measure theory the radonnikody. Before we prove this theorem, consider a simpler property. The measure of the cantor set is 1, while its lebesgue measure is 0. Gardiner and nikola lakic, quasiconformal teichmuller theory, 2000 75 greg hjorth, classification and orbit equivalence relations. Measuring things already the ancient greeks developed a theory of how to measure length, area, and volume and area of 1. Some results in probability theory require very deep measure theory. Courtesy of the mactutor history of mathematics archive, university of st.
Here are various notes on measure theory from a course at the university of melbourne. Greg hjorth 14 june 1963 january 2011 was an australian professor of mathematics, chess international master 1984 and joint with ian rogers commonwealth champion in 1983. Measure theory is the formal theory of things that are measurable. A measure on a topological space for which the measurable sets is the borel algebra bx is called a borel measure. Selected titles in this series 89 michel ledoux, the concentration of measure phenomenon. This barcode number lets you verify that youre getting exactly the right version or edition of a book. Measure theory 7 what is a lebesgue measurable subset. Greg hjorths 77 research works with 846 citations and 925 reads, including. Greg hjorths 77 research works with 8 citations and 922 reads, including. All such spaces are isomorphic to the unit interval with lebesgue measure. I have the generous permission by greg hjorth to use his notes on measure theory.
Algebra of sets 5 2 lebesgue measure on r 21 3 measurable functions 33 4 convergence a. Borel versus measure theoretic notions in the study of equivalence relations. In this introductory chapter we set forth some basic concepts of measure theory, which will open for abstract lebesgue integration. Introduction in the following we survey the main results in the theory of measure and integration. Gilbargtrudinger 5, geometric measure theory, geometric group theory, ergodic theory and dynamical systems, and di erential topology and geometry. B is a measure of difficulty of classification for different. On how to say the name hjorth in case you actually want to know. The name is danish, and in complete frankness most of my family long ago gave up the idea of persuading anyone to use the danish pronunciation, and instead have pronounced it as horth. Stroock, an introduction to the analysis of paths on a riemannian manifold, 2000 73 john locker, spectral theory of nonselfadjoint twopoint differential operators, 2000. Pages in category measure theory the following 155 pages are in this category, out of 155 total. Signed measures up until now our measures have always assumed values that were greater than or equal to 0. Some personal information, in case any old friends end up at this page while surfing late one night. Other terms are classical probability theory and measuretheoretic probability theory.
Analogously, a set in a measure space is said to have a. Borel measures play a preeminent role in measure theory on rn. The borelcantelli lemmas or the kolmogorov 01law are not hard to prove but hard to even state without measure theory. You can also watch all of marty rosss amsi lectures in 2006 by following. Turbulence and classification department of mathematics cornell. The connection between descriptive set theory and functional analysis can be traced back at least as far. In probability theory, the concentration of measure is a property of a large. Measurable sets 44 chapter lilt extension ok measures 12.
The main references i have used are eg, s and r, in that order. I hope that it will be found useful by undergraduates meeting lebesgue measure for the. Lebesgue integrals with such notion of measure, there is a corresponding integrability and integral, due to lebesgue. We show that there is a complete, consistent borel theory which has no borel. Measure and integration mathematics mit opencourseware. Measuring the classification difficulty of countable torsion. Measure theory mark dean lecture notes for fall 2015 phd class in decision theory brown university 1introduction next, we have an extremely rapid introduction to measure theory. Outer measures, premeasures, and product measures 179 chapter 2. Some personal information, in case any old friends end up at this page.
Distance on a set algebra defined by a subadditive set function 14. A measure theory tutorial measure theory for dummies. Lecture notes on measure theory and functional analysis. They include functional analysis and partial di erential equations see e.
Let me try to answer this by explaining how measure theory is used in wall street and why investment banks hire mathematicians. Measure theory together with x from an additive system on which is additive but not completely additive if x 2. Another possibility to build measures is their product. Greg hjorths research works university of melbourne.
Measure theory 6 outer measure of an interval is equal to its length. Descriptive set theory concerns the fine structure and regularity properties of definable subsets of polish spaces, e. Hutchinson department of mathematics school of mathematical sciences, a. Knowledge of first order logic is also useful, but not required. This is extremely important to probability because if we cant measure the probability of something then what good does all this work do us. Descriptive set theory, spring 2012 university of illinois. Greg hjorth was supported by an australian professorial fellowship from the australian research council. Measuring the classification difficulty of countable torsionfree abelian groups. However, ive come across measure theory several times, and i know it is a basis for probability theory, and, unsurprising, looking at a basic introduction such as this measure theory tutorial pdf, i see there are concepts such as events, sample spaces, and ways of getting from them to real numbers, that seem familiar.
Greg hjorth alain louveau descriptive set theory, spring 2012 instructor. Universal sets for pointsets properly on the nth level of the projective hierarchy. Lecture notes by alekos kechris and robin tuckerdrob pdf. Moreover, as formalized by kolmogorov 1933, measure theory provides the foundation of probability. Borel structures and borel theories article pdf available in journal of symbolic logic 762.
Greg hjorths research works university of melbourne, melbourne. Proofs are usually only sketched, but i have attempted to provide a. Selected titles in this series american mathematical society. A measure theory tutorial measure theory for dummies maya r. Greg hjorth s 77 research works with 8 citations and 922 reads, including. Then there are powerful methods that actually require measure theory. It amounts to replacing the literal rectangles used in riemann integration by more general rectangles, with bases not just intervals, but measurable sets, as follows. The pseudometric space defined by an outer measure 15. Gardiner and nikola lakic, quasiconformal teichmuller theory, 2000 75 greg hjorth, classification and orbit equivalence relations, 2000 74 daniel w. Okay, so what is measure theory all about luckily it is one of those wellnamed areas of mathematics.
Measure reducibility of countable borel equivalence relations. Greg hjorth in classification and orbit equivalence. Photograph of henri leon lebesgue 18751941, who laid the groundwork for modern measure theory and remade the theory of integration. Greg hjorth suddenly and unexpectedly passed away on january, 2011. The exterior measure, as the name indicates, attempts to describe. Borel versus measure theoretic notions in the study of. Greg hjorth, who was my dear friend, colleague, and ph.923 506 1262 904 735 473 321 619 1153 1220 185 712 363 499 947 1276 1228 547 372 328 1193 550 296 895 1179 799 838 1473 57 1350 1418 1457 268 381 1347